Starmaps seem so crystal pure and elegant. But reality is far more messy. The night sky seem to be decorated with static stars firmly nailed to the firmament, but reality is closer to an explosion in a glitter factory.
The only reason they seem to be stationary is because
Some planetarium computer programs have an option to show how distorted the constellations appear millions of years in the past ( as seen by the dinosaurs ) or millions of years in the future ( as seen by all the mutant rats and cockroaches ).
And in times gone by, a star that was quite close to another could now be quite remote. This can have implications for young upstart interstellar explorers who want to loot ancient high-tech goodies in the ruins of long gone empires. What used to be a nice compact empire half a million years ago is now spread out over a large section of the spiral arm.
But again I warn you, the math is not going to be easy.
Information about the motion of a star is generally given in terms of Radial Velocity and Proper Motion. Why? Because they are the easiest things for an astronomer to measure, that's why.
Any star's motion ( or vector ) is a complicated thing, but any vector can be split into two vectors, much in the same way that any line can be drawn using the two controls of an Etch-a-Sketch. So one knob will be the Radial Velocity knob, and the other the Proper Motion knob. The vector is called the Space Velocity.
In the diagram to the right, Vr is the Radial Velocity and Vt is the Proper Motion. Vs is the Space Velocity. Don't you just love all the weird terms astronomers invent?
Radial velocity is simply how fast the star is approaching or running away from the Sun ( the rate the RV knob is being turned ). It is easy to measure this with a spectroscope, as the star's spectrum is under the influence of the dreaded Doppler Effect. (I'm not going to attempt to explain this here, but five minutes with a web searcher should turn up a few tutorials). Most star's RV's are less than 100 kilometers per second, they are typically from 10 to 40 km/s. Positive values mean the star is receeding from the Sun, negative values are approaching.
Proper motion is how fast the star seems to move left or right in the sky, as viewed from the Sun (the rate of the PM knob is being turned). It is measured by taking photographs of stars from year to year and painstakingly measuring how much the image moves. The poor astronomer's task is complicated by the fact that all the stars are moving, so there isn't any reference point. But that's OK, that is what graduate students were invented for. What is generally given in a star catalog is the Annual Proper Motion, how many seconds of arc a star moves in one year. This is symbolized by the greek letter Mu μ. The largest known proper motion is Barnard's Star (often known as Barnard's Runaway Star) with an mu of 10.2 seconds per year.
The direction a star appears to be moving on the sky is called, surprise, surprise, the Direction of Proper Motion (sometimes called the Direction Angle of Proper Motion). It is in degrees. It is measured clockwise, and zero is north.
In the Gliese 3.0 data, Proper Motion is in the field of bytes 31-36, Radial Velocity is in bytes 44-49, and Direction Angle of Proper Motion is in bytes 38-42.
Here's a worked example using Barnard's star:
Variable Symbol Value Proper motion (RA direction) pmRA -0.798 arcsec/yr Proper motion (Dec direction) pmDec 10.327 arcsec/yr Radial velocity RV -111.0 km/s Parallax plx 0.5490 arcseconds Right Ascension (2000) RA 17 57.8 = 269.45 deg Declination (2000) Dec +04 41.6 = 4.69 deg
I'm still working on how to calculate this backwards (i.e., how to calculate what the constellation of the Big Dipper will look like two million years from now)
As a matter of interest, there is a star called Gliese 710 that is apparently on a near-collision course with the Sun. The current best estimate has it passing within a mere 0.4 parsecs about 1.4 million years from now. Gliese 710 is a type M1 red dwarf currently 19 parsecs (63 light years) away.